I imagine you will not be seeing her over Christmas, as you will both be with your families, so why not send her some flowers with a nice card that you personally penned.
Meaning, go to a flower shop with a sealed card in which you wrote something nice and personal. Hand it to the person and say, "I would like this card to go out with a dozen roses (or whatever) on Christmas Eve."
She will like it, and it won't freak her out.
Then take her out to a nice dinner on the 30th for her birthday, and buy her something small but nice. Maybe a book or a cd. Even something philanthropic like donating to something in her name. Stay away from jewelry or clothes until you know her style. It is too early to buy her a huge gift, you will make her feel awkward.
Don't make plans for the 31st, she might want to go out with her friends for New Year's Eve.
If you play it right on the 30th, she will be talking about you to all her friends the next night, and you will be a huge stud.